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The resolvent structure of a Volterra equation with nonsummable difference kernel

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Abstract

In this paper we study the asymptotic behavior of the resolvent of a Volterra linear integral equation whose difference kernel is nonsummable. For a certain class of such kernels the equation is reducible to an equation whose difference kernel is summable. This enables one to use the well-known results on the structure of resolvents of summable kernels in the case of a nonsummable kernel. We apply the obtained results to homogeneous kernels of degree s-1.

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Reference

  1. Z. B. Tsalyuk, “Asymptote of the Resolvent of the Volterra Equation with a Difference Kernel with Power Singularities of a Symbol,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 7, 77-84 (2006) [Russian Mathematics (Iz. VUZ) 50(7), 74–81 (2006)].

    MathSciNet  Google Scholar 

  2. V. A. Derbenev and Z. B. Tsalyuk, Asymptotes of Solutions to Linear Volterra Equations with Difference Kernels (KubGU, Krasnodar, 2001) [in Russian].

    Google Scholar 

  3. V. A. Derbenev and Z. B. Tsalyuk, “Asymptotic Behavior of the Resolvent of an Unstable Volterra Equation with Kernel Depending on the Difference of the Arguments,” Matem. Zametki 62(1), 88–94 (1997).

    MathSciNet  Google Scholar 

  4. J. J. Levin and D. F. Shea, “On the Asymptotic Behavior of the Bounded Solutions of Some Integral Equations.” I, J. Math. Anal. Appl. 37(1), 42–82 (1972).

    Article  MathSciNet  Google Scholar 

  5. J. J. Levin and D. F. Shea, “On the Asymptotic Behavior of the Bounded Solutions of Some Integral Equations.” II, J. Math. Anal. Appl. 37(2), 288–326 (1972).

    Article  MathSciNet  Google Scholar 

  6. J. J. Levin and D. F Shea, “On the Asymptotic Behavior of the Bounded Solutions of Some Integral Equations.” III, J. Math. Anal. Appl. 37(3), 537–575 (1972).

    Article  MathSciNet  Google Scholar 

  7. Z. B. Tsalyuk, “The Structure of the Resolvent of a Renewal Equation,” Izv. Vyssh. Uchebn. Zaved. Sev.-Kavkazsk. Region. Estestv. Nauki. Spetsvypusk, 150–151 (2001).

  8. V. A. Derbenev, “Asymptotic Properties of a Solution to Some Renewal Equation,” Matem. Analiz, Krasnodar, No. 2, 43–49 (1974).

    MathSciNet  Google Scholar 

  9. Z. B. Tsalyuk, “Admissibility of a Pair (Y, X) and Asymptotic Properties of the Resolvent for a System of Integral Volterra Equations,” Differents. Uravneniya 34(9), 1226–1230 (1998).

    MathSciNet  Google Scholar 

  10. Z. B. Tsalyuk and M. V Tsalyuk, “Asymptotics of the Resolvent of the Volterra Equation with a Nonintegrable Difference Kernel,” Differents. Uravneniya 39(6), 844–847 (2003).

    MathSciNet  Google Scholar 

  11. D. F. Shea and S. Wainger, “Variants of the Wiener-Levy Theorem with Applications to Stability Problems for Some Volterra Integral Equations,” Amer. J. Math. 97, 312–343 (1975).

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Kuban State University, ul. Stavropol’skaya 149, Krasnodar, 350040, Russia

    Z. B. Tsalyuk & M. B. Tsalyuk

Authors
  1. Z. B. Tsalyuk
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  2. M. B. Tsalyuk
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Correspondence to Z. B. Tsalyuk.

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Original Russian Text © Z.B. Tsalyuk and M.B. Tsalyuk, 2010, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2010, No. 4, pp. 72–82.

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Tsalyuk, Z.B., Tsalyuk, M.B. The resolvent structure of a Volterra equation with nonsummable difference kernel. Russ Math. 54, 62–71 (2010). https://doi.org/10.3103/S1066369X10040080

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  • DOI: https://doi.org/10.3103/S1066369X10040080

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